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We provide a new and much simpler structure for quasi-ideal adequatetransversals of abundant semigroups in terms of spined products, which issimilar in nature to that given by Saito for weakly multiplicative inversetransversals of regular semigroups. As a consequence we deduce a similar resultfor multiplicative transversals of abundant semigroups and also consider thecase when the semigroups are in fact regular and provide some new structuretheorems for inverse transversals.

Quasi-ideal transversals of abundant semigroups and spined products